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If it is 1, then n may be prime. If a n −1 (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, if a n −1 (modulo n) is 1, then n is usually prime. But here is a counterexample: if n = 341 and a = 2, then even though 341 = 11·31 is composite.
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. [ 1 ] [ 2 ] It is the basis of the Pratt certificate that gives a concise verification that n is prime.
There are infinitely many Fermat pseudoprimes to any given basis a > 1. [1]: Theorem 1 Even worse, there are infinitely many Carmichael numbers. [2] These are numbers for which all values of with (,) = are Fermat liars. For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors.
Input #1: b, the number of bits of the result Input #2: k, the number of rounds of testing to perform Output: a strong probable prime n while True: pick a random odd integer n in the range [2 b −1 , 2 b −1] if the Miller–Rabin test with inputs n and k returns “ probably prime ” then return n
The sieve starts with a list of the integers from 1 to n. From this list, all numbers of the form i + j + 2ij are removed, where i and j are positive integers such that 1 ≤ i ≤ j and i + j + 2ij ≤ n. The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (that is, all primes except 2) below 2n + 2.
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...
Optionally, perform trial division to check if n is divisible by a small prime number less than some convenient limit. Perform a base 2 strong probable prime test. If n is not a strong probable prime base 2, then n is composite; quit. Find the first D in the sequence 5, −7, 9, −11, 13, −15, ... for which the Jacobi symbol (D/n) is −1.